# Why does the sum of all natural numbers equal -1/12? **URL:** \ **Category:** Factual Questions\ **Created:** [January 10, 2014, 5:58pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452 "2014-01-10T17:58:53Z")\ **Posts on this page:** 20\ **Page:** 7
**Author:** ![wevets](https://avatars.discourse-cdn.com/v4/letter/w/e9a140/32.png) [@wevets](https://boards.straightdope.com/u/wevets)\ **Post date:** [January 18, 2014, 3:29am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/121 "2014-01-18T03:29:19Z")
@ **Indistinguishable** & @ **Asympotically fat** I think I get it … so would it be fair to say that -1/12 is **an** answer to the summation, not necessarily **the only** answer? Thanks - sorry for dropping in not having read the thread, but I saw the Chu-Carroll post and realized I’d seen the same thing on the GQ 1st page for several days now. 🙂 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 18, 2014, 6:03am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/122 "2014-01-18T06:03:57Z")
> [@wevets](#): > > @ **Indistinguishable** & @ **Asympotically fat** I think I get it … so would it be fair to say that -1/12 is **an** answer to the summation, not necessarily **the only** answer? Yes, but it’s a reasonable answer in the sense that the methods that we use to assign that sum have properties that we consider desirable. If you want to say that it’s something else, you need to explain how you’re making that assignment, and you need to show that the methods that you’re using have equally nice properties. I’m not aware of any divergent series that is assigned multiple values by reasonable notions of summation. Anyone? ---
**Author:** ![wevets](https://avatars.discourse-cdn.com/v4/letter/w/e9a140/32.png) [@wevets](https://boards.straightdope.com/u/wevets)\ **Post date:** [January 18, 2014, 8:28am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/123 "2014-01-18T08:28:04Z")
Thanks, **ultrafilter** - I wasn’t really wondering is another value specifically would make a good answer, but whether ‘undefined’ is just as much an answer as ‘-1/12’ is… (I’m guessing yes?) ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [January 18, 2014, 9:52am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/124 "2014-01-18T09:52:57Z")
> [@Asympotically\_fat](#): > > I remember when I got my first programmable calculator, it had a program that would quickly solve quadratic equations, but it also had a program that would ask you to type in your name and would return “Hello \, you are a wanker”. The lesson here being that we shouldn’t take everything calculators say to heart. That’s probably my favourite post in this thread. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [January 18, 2014, 10:24am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/125 "2014-01-18T10:24:23Z")
Damn, leafing through those linked blogs and their comment threads makes me somewhat despair of the quality of discourse on the net, especially given that they are all part of the so-called ‘sceptical’ community that prides itself in rationality… ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [January 18, 2014, 1:40pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/126 "2014-01-18T13:40:33Z")
> [@Half\_Man\_Half\_Wit](#): > > That’s probably my favourite post in this thread. But maybe **Asympotically fat** and his friends _were_ all wankers. Then his post doesn’t prove anything. 🙂 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 18, 2014, 8:08pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/127 "2014-01-18T20:08:07Z")
> [@wevets](#): > > Thanks, **ultrafilter** - I wasn’t really wondering is another value specifically would make a good answer, but whether ‘undefined’ is just as much an answer as ‘-1/12’ is… (I’m guessing yes?) You could just insist that divergent series don’t have sums, but in doing so you’d be throwing out a tool that is useful in making predictions about actual physical systems (albeit quantum ones). I’ve been looking for some information on the specifics of how divergent series are used in quantum mechanics, and I found [a very nice explanation](http://physics.stackexchange.com/questions/93124/divergent-series) of a modern mathematician’s view of divergent series. In particular, I want to quote this bit: > [@](#): > > As G. H. Hardy’s “Divergent Series” explain in page 6, the trick is to understand that our usual notion of sum of a serie is a way to define something we call a “sum”. In other words given a sequence we have a map that atributes to this sequence a number. The “sum map” being the trivial operation of summing the terms if the series is absolutely convergent. The idea behind divergent series is to realize that this map althogh in a sense canonical is not unique. ---
**Author:** ![wevets](https://avatars.discourse-cdn.com/v4/letter/w/e9a140/32.png) [@wevets](https://boards.straightdope.com/u/wevets)\ **Post date:** [January 18, 2014, 11:49pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/128 "2014-01-18T23:49:44Z")
Ah! That helps explain a bit. Thanks, **ultrafilter**! 🙂 ---
**Author:** ![JohnClay](https://avatars.discourse-cdn.com/v4/letter/j/c37758/32.png) [@JohnClay](https://boards.straightdope.com/u/JohnClay)\ **Post date:** [January 20, 2014, 3:58am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/129 "2014-01-20T03:58:06Z")
BTW in case this hasn’t been mentioned before… In some kind of sum, 1 + 2 + 4 + 8 + 16 +… = -1 > **[1 + 2 + 4 + 8 + ⋯](https://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF)** > > In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity. > However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example,... ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 30, 2014, 7:57pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/130 "2014-01-30T19:57:00Z")
Ack, I’ve been failing to find time to keep up with this thread, but … > [@Half\_Man\_Half\_Wit](#): > > Damn, leafing through those linked blogs and their comment threads makes me somewhat despair of the quality of discourse on the net, especially given that they are all part of the so-called ‘sceptical’ community that prides itself in rationality… Yeah, exactly. One thing I wanted to note, way back when, was that I was quite heartened by this thread; the SDMB seemed to be the one place on the Internet where people by and large really _got it_, so to speak (the vast majority of the discussion elsewhere, even from people one might expect better from, being terribly, terribly disappointing, as you note). ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [January 31, 2014, 1:49am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/131 "2014-01-31T01:49:37Z")
Back in [this post](http://boards.straightdope.com/sdmb/showpost.php?p=16424375&postcount=159) in an earlier thread, **Indistinguishable** wrote > [@](#): > > Given a series a1 + a2 + a3 + a4 + …, we can form the associated function f(h) = a1 e-1h + a2 e-2h + a3 e-3h + a4 e-4h + …, such that the desired sum is the value f(0). Using exp(-n_h) seems arbitrary. We could use a different regularization function in place of exp(n_h), like 1/(1+n\*h). Then we get f(h) = a1/(1+1h) + a2/(1+2h) + a3/(1+3h) + a4/(1+4h) + … Do we still get 1 - 1 + 1 - 1 + … = 1/2? Playing around with it numerically, that looks to be the case. If I were willing to get a pen and piece of paper, I could probably show that it does. Since 1/(1+n_h) decays less rapidly than exp(-n_h), there are going to be series which converge using the later function, but not using the former, so you could perhaps argue that exp(-n_h) is “better”. But if our series we want to sum diverges too rapidly, exp(n_h) won’t decay rapidly enough either, and we’d _have_ to pick a different regularization function to get a value. i don’t think you can find one function that will always work. There are many different definitions of integration. I read somewhere over a hundred. They don’t all work all the time. Some functions aren’t integrable using one definition, but are integrable using a different one. But (as far as I know (and correct me if I’m wrong!)) all the definitions that can integrate some function agree on the value for that integral. Similarly, there are an unlimited number of different regularization functions one could use for a given series. Can it be shown that the ones which converge to a finite value all give the same answer for that series? Or can someone give a counter-example? (No pathological ones, like 0\*h, and all integrals equal zero.) I guess I’m thinking here mostly in terms of series whose partial sums alternate in sign, diverging in magnitude, but not just towards +infinity or just towards -infinity. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2014, 9:53am UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/132 "2014-01-31T09:53:03Z")
Yes, you will get the same results for series like 1 - 1 + 1 - 1 + … and 1 - 2 + 3 - 4 + … using other decay functions (including the one you chose). As noted above, it’s convenient to define Abel summation using exponential decay, but > [@Indistinguishable](#): > > (actually, the exponentiality doesn’t matter, and we would get the same results using any sufficiently smooth decay function, but I’ll leave that discussion for later…) Using exponential decay amounts to thinking in terms of Fourier transforms; we are then essentially taking the sum of a function to be the limiting value of its Fourier transform at 0. Actually, you can use the perspective of Fourier theory more generally, to understand the “any sufficiently smooth decay function” result; handwavily, as the decay coefficient sequence d[sub]0[/sub], d[sub]1[/sub], d[sub]2[/sub], … moves towards constantly 1 (no decay), its Fourier transform “moves towards the Fourier transform of the constantly 1 function (i.e., the Dirac delta)”, so that the sum using such a decay is equal to the convolution of the Fourier transform of the series being summed (i.e., the associated function as we’ve defined it above using exponential decay (slightly rotated)) with a function “moving towards the Dirac delta”, which amounts to the limiting value of this associated function at 0. (To make this argument analytically rigorous, we need to clarify our understanding of “moves towards the Dirac delta”; the smoothness requirements will come in here.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2014, 10:51pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/133 "2014-01-31T22:51:18Z")
> [@Indistinguishable](#): > > …so that the sum using such a decay is equal to **the value at 0 of** the convolution of the Fourier transform of the series being summed… Missing words reinstated in bold. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [February 1, 2014, 8:01pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/134 "2014-02-01T20:01:55Z")
(Perhaps [this post](http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/) by Terence Tao will be of use in clarifying the universality properties of smoothed sums) ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [February 2, 2014, 6:09pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/135 "2014-02-02T18:09:27Z")
> [@Chronos](#): > > But then again, the plus sign in “2+3=5” isn’t the same plus sign as the one in “2.3+3.4=5.7”, either, but I’m guessing that doesn’t bother you. Yeah, they are. You’re using the one for real numbers, which gives the same answers as for the ones for integers on the domain of integers. I’d say the addition of real numbers is what most assume when they see the plus sign. I’d include complex numbers, but I’m not entirely sure you can say that ordered pairs that cannot be reduced really count as addition. If so, then throw those in. The point is that people do not in any way think it involves extending some function. Addition is assumed to be one of the simplest operations, and thus can’t consist of all the operations needed to do all these other things. People can’t envision getting from finite real summation to Abel summation as easily as they can get from finite real summation to infinite limit-based summation. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 3, 2014, 4:29pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/136 "2014-02-03T16:29:24Z")
No, the plus sign for real numbers can’t be applied to the domain of integers. It’s true that 2.0+3.0=5.0, and that 2+3=5, but those are two different statements, because 2 is not the same object as 2.0, nor is 3 the same object as 3.0, nor 5 the same as 5.0 . Now, in actual practice, this is often glossed over, because there is a very strong sort of correspondence between the objects 2 and 2.0 , 3 and 3.0 , and so on ( **Indistinguishable** can tell us the name of that correspondence; I always get the something-morphisms mixed up). And so, given that the real-domain plus works the same way on those corresponding objects that the integer-domain plus works on integers, we can usually get away with pretending that the integer-domain plus is a special case of the real-domain plus, even though it isn’t. But by the same token, we can similarly get away with pretending that the sum of a convergent series is a special case of the sum of various sorts of more general series. ---
**Author:** ![clarkcompanero](https://avatars.discourse-cdn.com/v4/letter/c/76d3ee/32.png) [@clarkcompanero](https://boards.straightdope.com/u/clarkcompanero)\ **Post date:** [March 13, 2015, 4:52pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/137 "2015-03-13T16:52:01Z")
> [@standingwave](#): > > OK, Numberphile just blew my mind. The sum of all natural numbers (from 1 to infinity) produces an “astounding” result. > > [http://youtu.be/w-I6XTVZXww](http://youtu.be/w-I6XTVZXww) > > I can’t help feeling that it’s a bit of a parlor trick like “proving” 1 = 2 via mathematical trickery. All that sliding infinite series one spot to the right or left to make things cancel. Something interesting as well. Take any number, any number at all and half that number, then half the answer ( 10,000,5,000,2500,1250,625,312.50…………) all the way to infinity (I used excel). If you add all those numbers up, it comes out to exactly double the original starting number. So if I star with 10 million and continue dividing by 2 until infinity, and add them all up, I will get 20 million ---
**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\ **Post date:** [March 13, 2015, 4:58pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/138 "2015-03-13T16:58:44Z")
The series you describe will approach twice the original value, but never quite reach it. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 13, 2015, 5:10pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/139 "2015-03-13T17:10:02Z")
> [@D18](#): > > The series you describe will approach twice the original value, but never quite reach it. He specified “all the way to infinity.” The sum of the infinite series is exactly 2n, in exactly the same way that 0.999999… is exactly 1. ---
**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\ **Post date:** [March 13, 2015, 5:13pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/140 "2015-03-13T17:13:58Z")
Not that I know what I’m talking about, but although he said “all the way to infinity” he said he demonstrated it with Excel, which has got to be a bit short of infinity, no? At any rate, I’ll concede your point, the Excel spreadsheet notwithstanding. [Previous page](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452.md?page=6) [Next page](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452.md?page=8)